Introduction to the General operation of Matrices

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Abstract: - The General operation of Matrices is an original study introduced by the author in the mathematical domain. The basic idea of the General operationΒ ...
Advances in Mathematical and Computational Methods

Introduction to the General operation of Matrices CLAUDE ZIAD BAYEH1, 2 1

Faculty of Engineering II, Lebanese University 2 EGRDI transaction on mathematics (2004) LEBANON Email: [email protected]

NIKOS E.MASTORAKIS WSEAS (Research and Development Department) http://www.worldses.org/research/index.html Agiou Ioannou Theologou 17-2315773, Zografou, Athens,GREECE [email protected] Abstract: - The General operation of Matrices is an original study introduced by the author in the mathematical domain. The basic idea of the General operation of Matrices is similar to the traditional operation of matrices, but moreover the General operation of Matrices is more general and contains many forms and ways of multiplying, adding, subtracting and dividing many matrices. The main goal of introducing the General operation of Matrices is to facilitate the writing of many complicated equations in simple matrices. In this paper a brief study is introduced with the definition of the General operation of Matrices and simple applications are developed in order to give idea about the importance of this new concept of operation between Matrices. Many studies will follow this one in order to find more applications in mathematics and all scientific domains. Key-words:- General operation of Matrices, Matrix, Addition, Multiplication, Subtracting, Dividing, Relation between matrices.

applications in most scientific fields. In physics, matrices are used to study electrical circuits, optics, and quantum mechanics [10-11]. In this paper, the author introduced the General operation of Matrices which is the general case of the traditional operation between matrices. The concept of manipulating the General operation of Matrices is different from the traditional operation between matrices. The general operation of matrices is introduced in order to facilitate the writing of many complicated expressions into matrices and to facilitate the operation between matrices as we want. We can operate the Columns of the first matrix by the columns of the second matrix, or the columns of the first matrix by the Lines (rows) of the second matrix, or Lines of the first matrix by the line of the second matrix, or Lines of the first matrix by the columns of the second matrix. This is not the case of the traditional operation between matrices. Mainly we have two operations between matrices, the first operation is between the elements of the two matrices and the second is the operation between the operated elements. For example: (π‘Žπ‘Ž (𝑂𝑂𝑂𝑂1 )𝑏𝑏) is the first operation between two elements of the two matrices, with (𝑂𝑂𝑂𝑂1 ) can be replaced by an operator (+, -, *, /), the second operator (𝑂𝑂𝑂𝑂2 ) is the operator

1 Introduction In mathematics, a matrix (plural matrices) is a rectangular array of numbers, symbols, or expressions [1-2]. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements is 1 4 βˆ’2 οΏ½ οΏ½ 9 8 4 Matrices of the same size can be added or subtracted element by element [3-4]. The rule for matrix multiplication is more complicated, and two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second [5-7]. A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x. For example, the rotation of vectors in three dimensional space is a linear transformation [8-9]. The product of two matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of a system of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Eigenvalues and eigenvectors provide insight into the geometry of linear transformations. Matrices find

ISBN: 978-1-61804-117-3

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between the operated elements such as (π‘Žπ‘Ž 𝑂𝑂𝑂𝑂1 𝑏𝑏)𝑂𝑂𝑂𝑂2 (𝑐𝑐 𝑂𝑂𝑂𝑂1 𝑑𝑑). With (𝑂𝑂𝑂𝑂2 ) can be replaced by an operator (+, -, *, /). Many studies will follow this paper in order to find other applications in many domains as in science, engineering and mathematics. In the second section, a definition of the general operation of matrices is presented. in the section 3, practical examples of the General operation of matrices are presented. and finally, a conclusion is presented in the section 4.

𝐢𝐢𝐢𝐢 𝐢𝐢𝐢𝐢 𝑒𝑒 𝑓𝑓 π‘Žπ‘Ž 𝑏𝑏 οΏ½ �𝑂𝑂𝑂𝑂1 οΏ½ οΏ½ οΏ½ β€’ 𝐴𝐴 �𝑂𝑂𝑂𝑂1 οΏ½ 𝐡𝐡 = οΏ½ 𝑔𝑔 β„Ž 𝑐𝑐 𝑑𝑑 𝑂𝑂𝑂𝑂 𝑂𝑂𝑂𝑂2 2 = (π‘Žπ‘Ž 𝑂𝑂𝑂𝑂1 𝑒𝑒)𝑂𝑂𝑂𝑂2 (𝑐𝑐 𝑂𝑂𝑂𝑂1 𝑓𝑓) (𝑏𝑏 𝑂𝑂𝑂𝑂1 𝑒𝑒)𝑂𝑂𝑂𝑂2 (𝑑𝑑 𝑂𝑂𝑂𝑂1 𝑓𝑓) οΏ½ οΏ½ (π‘Žπ‘Ž 𝑂𝑂𝑂𝑂1 𝑔𝑔)𝑂𝑂𝑂𝑂2 (𝑐𝑐 𝑂𝑂𝑂𝑂1 β„Ž) (𝑏𝑏 𝑂𝑂𝑂𝑂1 𝑔𝑔)𝑂𝑂𝑂𝑂2 (𝑑𝑑 𝑂𝑂𝑂𝑂1 β„Ž)

2 Definition of the General operation of matrices

𝐿𝐿𝐿𝐿 𝐿𝐿𝐿𝐿 𝑒𝑒 𝑓𝑓 π‘Žπ‘Ž 𝑏𝑏 𝑂𝑂𝑂𝑂 οΏ½ οΏ½ 1οΏ½ οΏ½ οΏ½ β€’ 𝐴𝐴 �𝑂𝑂𝑂𝑂1 οΏ½ 𝐡𝐡 = οΏ½ 𝑔𝑔 β„Ž 𝑐𝑐 𝑑𝑑 𝑂𝑂𝑂𝑂 𝑂𝑂𝑂𝑂2 2 (π‘Žπ‘Ž 𝑂𝑂𝑂𝑂1 𝑒𝑒)𝑂𝑂𝑂𝑂2 (𝑏𝑏 𝑂𝑂𝑂𝑂1 𝑔𝑔) (π‘Žπ‘Ž 𝑂𝑂𝑂𝑂1 𝑓𝑓)𝑂𝑂𝑂𝑂2 (𝑏𝑏 𝑂𝑂𝑂𝑂1 β„Ž) οΏ½ =οΏ½ (𝑐𝑐 𝑂𝑂𝑂𝑂1 𝑒𝑒)𝑂𝑂𝑂𝑂2 (𝑑𝑑 𝑂𝑂𝑂𝑂1 𝑔𝑔) (𝑐𝑐 𝑂𝑂𝑂𝑂1 𝑓𝑓)𝑂𝑂𝑂𝑂2 (𝑑𝑑 𝑂𝑂𝑂𝑂1 β„Ž)

𝐿𝐿𝐿𝐿 𝐿𝐿𝐿𝐿 𝑒𝑒 𝑓𝑓 π‘Žπ‘Ž 𝑏𝑏 𝑂𝑂𝑂𝑂 οΏ½ οΏ½ 1οΏ½ οΏ½ οΏ½ β€’ 𝐴𝐴 �𝑂𝑂𝑂𝑂1 οΏ½ 𝐡𝐡 = οΏ½ 𝑔𝑔 β„Ž 𝑐𝑐 𝑑𝑑 𝑂𝑂𝑂𝑂 𝑂𝑂𝑂𝑂2 2 (π‘Žπ‘Ž 𝑂𝑂𝑂𝑂1 𝑒𝑒)𝑂𝑂𝑂𝑂2 (𝑏𝑏 𝑂𝑂𝑂𝑂1 𝑓𝑓) (π‘Žπ‘Ž 𝑂𝑂𝑂𝑂1 𝑔𝑔)𝑂𝑂𝑂𝑂2 (𝑏𝑏 𝑂𝑂𝑂𝑂1 β„Ž) οΏ½ =οΏ½ (𝑐𝑐 𝑂𝑂𝑂𝑂1 𝑒𝑒)𝑂𝑂𝑂𝑂2 (𝑑𝑑 𝑂𝑂𝑂𝑂1 𝑓𝑓) (𝑐𝑐 𝑂𝑂𝑂𝑂1 𝑔𝑔)𝑂𝑂𝑂𝑂2 (𝑑𝑑 𝑂𝑂𝑂𝑂1 β„Ž)

The general operation of matrices is denoted by: 𝑀𝑀1 𝑀𝑀2 οΏ½ 𝑂𝑂𝑂𝑂1 οΏ½ (1) 𝑂𝑂𝑂𝑂2

2.1 Examples of the general operation of matrices

With β€’ 𝑀𝑀1 𝑀𝑀2 is the form of operation between two matrices for example it is written as following: -𝐢𝐢𝐢𝐢 means the operation between the Columns of the first matrix and the Columns of the second matrix. -𝐢𝐢𝐢𝐢 means the operation between the Columns of the first matrix and the Lines of the second matrix. -𝐿𝐿𝐿𝐿 means the operation between the Lines of the first matrix and the Lines of the second matrix. -𝐿𝐿𝐿𝐿 means the operation between the Lines of the first matrix and the Columns of the second matrix.

In this section, we give some examples in order understand the principle of the general operation matrices and how it works. It is very important understand the basis of operation in order understand the whole operations.

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Suppose the following matrices: 𝑒𝑒 𝑓𝑓 π‘Žπ‘Ž 𝑏𝑏 οΏ½ and 𝐡𝐡 = οΏ½ οΏ½ 𝐴𝐴 = οΏ½ 𝑔𝑔 β„Ž 𝑐𝑐 𝑑𝑑

β€’ 𝑂𝑂𝑂𝑂1, is the first operation between the elements of the first and second matrices. It can be replaced by (+, βˆ’,Γ— π‘œπ‘œπ‘œπ‘œ Γ·).

β€’ 𝑂𝑂𝑂𝑂2 , is the secondary operation between the elements of the first and second matrices. It can be replaced by (+, βˆ’,Γ— π‘œπ‘œπ‘œπ‘œ Γ·).

2.1.1 The first operation is β€œ*” and the second is β€œ-” 𝐢𝐢𝐢𝐢 𝐢𝐢𝐢𝐢 𝑒𝑒 𝑓𝑓 π‘Žπ‘Ž 𝑏𝑏 οΏ½οΏ½ βˆ— οΏ½οΏ½ οΏ½ β€’ 𝐴𝐴 οΏ½ βˆ— οΏ½ 𝐡𝐡 = οΏ½ 𝑔𝑔 β„Ž 𝑐𝑐 𝑑𝑑 βˆ’ βˆ’ (π‘Žπ‘Ž βˆ— 𝑒𝑒) βˆ’ (𝑐𝑐 βˆ— 𝑔𝑔) (𝑏𝑏 βˆ— 𝑒𝑒) βˆ’ (𝑑𝑑 βˆ— 𝑔𝑔) οΏ½ =οΏ½ (π‘Žπ‘Ž βˆ— 𝑓𝑓) βˆ’ (𝑐𝑐 βˆ— β„Ž) (𝑏𝑏 βˆ— 𝑓𝑓) βˆ’ (𝑑𝑑 βˆ— β„Ž)

𝐢𝐢𝐢𝐢 𝐢𝐢𝐢𝐢 𝑒𝑒 𝑓𝑓 π‘Žπ‘Ž 𝑏𝑏 𝑂𝑂𝑂𝑂 οΏ½ οΏ½ 1οΏ½ οΏ½ οΏ½ β€’ 𝐴𝐴 οΏ½ 𝑂𝑂𝑂𝑂1 οΏ½ 𝐡𝐡 = οΏ½ 𝑔𝑔 β„Ž 𝑐𝑐 𝑑𝑑 𝑂𝑂𝑂𝑂2 𝑂𝑂𝑂𝑂2 = (π‘Žπ‘Ž 𝑂𝑂𝑂𝑂1 𝑒𝑒)𝑂𝑂𝑂𝑂2 (𝑐𝑐 𝑂𝑂𝑂𝑂1 𝑔𝑔) (𝑏𝑏 𝑂𝑂𝑂𝑂1 𝑒𝑒)𝑂𝑂𝑂𝑂2 (𝑑𝑑 𝑂𝑂𝑂𝑂1 𝑔𝑔) οΏ½ οΏ½ (π‘Žπ‘Ž 𝑂𝑂𝑂𝑂1 𝑓𝑓)𝑂𝑂𝑂𝑂2 (𝑐𝑐 𝑂𝑂𝑂𝑂1 β„Ž) (𝑏𝑏 𝑂𝑂𝑂𝑂1 𝑓𝑓)𝑂𝑂𝑂𝑂2 (𝑑𝑑 𝑂𝑂𝑂𝑂1 β„Ž)

𝐿𝐿𝐿𝐿 𝐿𝐿𝐿𝐿 𝑒𝑒 𝑓𝑓 π‘Žπ‘Ž 𝑏𝑏 οΏ½οΏ½ βˆ— οΏ½οΏ½ οΏ½ β€’ 𝐴𝐴 οΏ½ βˆ— οΏ½ 𝐡𝐡 = οΏ½ 𝑔𝑔 β„Ž 𝑐𝑐 𝑑𝑑 βˆ’ βˆ’ (π‘Žπ‘Ž βˆ— 𝑒𝑒) βˆ’ (𝑏𝑏 βˆ— 𝑓𝑓) (π‘Žπ‘Ž βˆ— 𝑔𝑔) βˆ’ (𝑏𝑏 βˆ— β„Ž) οΏ½ =οΏ½ (𝑐𝑐 βˆ— 𝑒𝑒) βˆ’ (𝑑𝑑 βˆ— 𝑓𝑓) (𝑐𝑐 βˆ— 𝑔𝑔) βˆ’ (𝑑𝑑 βˆ— β„Ž)

Examples: Suppose the following matrices: 𝑒𝑒 𝑓𝑓 π‘Žπ‘Ž 𝑏𝑏 οΏ½ and 𝐡𝐡 = οΏ½ οΏ½ 𝐴𝐴 = οΏ½ 𝑔𝑔 β„Ž 𝑐𝑐 𝑑𝑑

ISBN: 978-1-61804-117-3

𝐢𝐢𝐢𝐢 𝐢𝐢𝐢𝐢 𝑒𝑒 𝑓𝑓 π‘Žπ‘Ž 𝑏𝑏 οΏ½ οΏ½ οΏ½ β€’ 𝐴𝐴 οΏ½ βˆ— οΏ½ 𝐡𝐡 = οΏ½ βˆ— οΏ½οΏ½ 𝑔𝑔 β„Ž 𝑐𝑐 𝑑𝑑 βˆ’ βˆ’ (π‘Žπ‘Ž βˆ— 𝑒𝑒) βˆ’ (𝑐𝑐 βˆ— 𝑓𝑓) (𝑏𝑏 βˆ— 𝑒𝑒) βˆ’ (𝑑𝑑 βˆ— 𝑓𝑓) οΏ½ =οΏ½ (π‘Žπ‘Ž βˆ— 𝑔𝑔) βˆ’ (𝑐𝑐 βˆ— β„Ž) (𝑏𝑏 βˆ— 𝑔𝑔) βˆ’ (𝑑𝑑 βˆ— β„Ž)

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𝐿𝐿𝐿𝐿 𝐿𝐿𝐿𝐿 𝑒𝑒 𝑓𝑓 π‘Žπ‘Ž 𝑏𝑏 οΏ½οΏ½+οΏ½οΏ½ οΏ½ β€’ 𝐴𝐴 οΏ½ + οΏ½ 𝐡𝐡 = οΏ½ 𝑔𝑔 β„Ž 𝑐𝑐 𝑑𝑑 / / (π‘Žπ‘Ž + 𝑒𝑒)/(𝑏𝑏 + 𝑓𝑓) (π‘Žπ‘Ž + 𝑔𝑔)/(𝑏𝑏 + β„Ž) οΏ½ =οΏ½ (𝑐𝑐 + 𝑒𝑒)/(𝑑𝑑 + 𝑓𝑓) (𝑐𝑐 + 𝑔𝑔)/(𝑑𝑑 + β„Ž)

𝐿𝐿𝐿𝐿 𝐿𝐿𝐿𝐿 𝑒𝑒 𝑓𝑓 π‘Žπ‘Ž 𝑏𝑏 οΏ½οΏ½ βˆ— οΏ½οΏ½ οΏ½ β€’ 𝐴𝐴 οΏ½ βˆ— οΏ½ 𝐡𝐡 = οΏ½ 𝑔𝑔 β„Ž 𝑐𝑐 𝑑𝑑 βˆ’ βˆ’ (π‘Žπ‘Ž βˆ— 𝑒𝑒) βˆ’ (𝑏𝑏 βˆ— 𝑔𝑔) (π‘Žπ‘Ž βˆ— 𝑓𝑓) βˆ’ (𝑏𝑏 βˆ— β„Ž) οΏ½ =οΏ½ (𝑐𝑐 βˆ— 𝑒𝑒) βˆ’ (𝑑𝑑 βˆ— 𝑔𝑔) (𝑐𝑐 βˆ— 𝑓𝑓) βˆ’ (𝑑𝑑 βˆ— β„Ž)

𝐿𝐿𝐿𝐿 𝐿𝐿𝐿𝐿 𝑒𝑒 𝑓𝑓 π‘Žπ‘Ž 𝑏𝑏 οΏ½οΏ½ + οΏ½οΏ½ οΏ½ β€’ 𝐴𝐴 οΏ½ + οΏ½ 𝐡𝐡 = οΏ½ 𝑔𝑔 β„Ž 𝑐𝑐 𝑑𝑑 / / (π‘Žπ‘Ž + 𝑒𝑒)/(𝑏𝑏 + 𝑔𝑔) (π‘Žπ‘Ž + 𝑓𝑓)/(𝑏𝑏 + β„Ž) οΏ½ =οΏ½ (𝑐𝑐 + 𝑒𝑒)/(𝑑𝑑 + 𝑔𝑔) (𝑐𝑐 + 𝑓𝑓)/(𝑑𝑑 + β„Ž)

2.1.2 The first operation is β€œ*” and the second is β€œ+” 𝐢𝐢𝐢𝐢 𝐢𝐢𝐢𝐢 𝑒𝑒 𝑓𝑓 π‘Žπ‘Ž 𝑏𝑏 οΏ½οΏ½ βˆ— οΏ½οΏ½ οΏ½ β€’ 𝐴𝐴 οΏ½ βˆ— οΏ½ 𝐡𝐡 = οΏ½ 𝑔𝑔 β„Ž 𝑐𝑐 𝑑𝑑 + + (π‘Žπ‘Ž βˆ— 𝑒𝑒) + (𝑐𝑐 βˆ— 𝑔𝑔) (𝑏𝑏 βˆ— 𝑒𝑒) + (𝑑𝑑 βˆ— 𝑔𝑔) οΏ½ =οΏ½ (π‘Žπ‘Ž βˆ— 𝑓𝑓) + (𝑐𝑐 βˆ— β„Ž) (𝑏𝑏 βˆ— 𝑓𝑓) + (𝑑𝑑 βˆ— β„Ž)

2.1.4 The first operation is β€œ*” and the second is β€œ-” with two rectangular matrices (3 columns and 2 lines) Suppose the following matrices: 𝑔𝑔 β„Ž 𝑖𝑖 π‘Žπ‘Ž 𝑏𝑏 𝑐𝑐 οΏ½ and 𝐡𝐡 = οΏ½ οΏ½ 𝐴𝐴 = οΏ½ 𝑑𝑑 𝑒𝑒 𝑓𝑓 𝑗𝑗 π‘˜π‘˜ 𝑙𝑙

𝐢𝐢𝐢𝐢 𝐢𝐢𝐢𝐢 𝑒𝑒 𝑓𝑓 π‘Žπ‘Ž 𝑏𝑏 οΏ½οΏ½ βˆ— οΏ½οΏ½ οΏ½ β€’ 𝐴𝐴 οΏ½ βˆ— οΏ½ 𝐡𝐡 = οΏ½ 𝑔𝑔 β„Ž 𝑐𝑐 𝑑𝑑 + + (π‘Žπ‘Ž βˆ— 𝑒𝑒) + (𝑐𝑐 βˆ— 𝑓𝑓) (𝑏𝑏 βˆ— 𝑒𝑒) + (𝑑𝑑 βˆ— 𝑓𝑓) οΏ½ =οΏ½ (π‘Žπ‘Ž βˆ— 𝑔𝑔) + (𝑐𝑐 βˆ— β„Ž) (𝑏𝑏 βˆ— 𝑔𝑔) + (𝑑𝑑 βˆ— β„Ž)

𝐢𝐢𝐢𝐢 π‘Žπ‘Ž β€’ 𝐴𝐴 οΏ½ βˆ— οΏ½ 𝐡𝐡 = οΏ½ 𝑑𝑑 βˆ’

𝐿𝐿𝐿𝐿 𝐿𝐿𝐿𝐿 𝑒𝑒 𝑓𝑓 π‘Žπ‘Ž 𝑏𝑏 οΏ½οΏ½ βˆ— οΏ½οΏ½ οΏ½ β€’ 𝐴𝐴 οΏ½ βˆ— οΏ½ 𝐡𝐡 = οΏ½ 𝑔𝑔 β„Ž 𝑐𝑐 𝑑𝑑 + + (π‘Žπ‘Ž βˆ— 𝑒𝑒) + (𝑏𝑏 βˆ— 𝑓𝑓) (π‘Žπ‘Ž βˆ— 𝑔𝑔) + (𝑏𝑏 βˆ— β„Ž) οΏ½ =οΏ½ (𝑐𝑐 βˆ— 𝑒𝑒) + (𝑑𝑑 βˆ— 𝑓𝑓) (𝑐𝑐 βˆ— 𝑔𝑔) + (𝑑𝑑 βˆ— β„Ž)

𝐿𝐿𝐿𝐿 𝐿𝐿𝐿𝐿 𝑒𝑒 𝑓𝑓 π‘Žπ‘Ž 𝑏𝑏 οΏ½οΏ½ βˆ— οΏ½οΏ½ οΏ½ β€’ 𝐴𝐴 οΏ½ βˆ— οΏ½ 𝐡𝐡 = οΏ½ 𝑔𝑔 β„Ž 𝑐𝑐 𝑑𝑑 + + (π‘Žπ‘Ž βˆ— 𝑒𝑒) + (𝑏𝑏 βˆ— 𝑔𝑔) (π‘Žπ‘Ž βˆ— 𝑓𝑓) + (𝑏𝑏 βˆ— β„Ž) οΏ½ =οΏ½ (𝑐𝑐 βˆ— 𝑒𝑒) + (𝑑𝑑 βˆ— 𝑔𝑔) (𝑐𝑐 βˆ— 𝑓𝑓) + (𝑑𝑑 βˆ— β„Ž) This equation is similar to the traditional operation between two matrices which gives the same result as following 𝑒𝑒 𝑓𝑓 π‘Žπ‘Ž 𝑏𝑏 οΏ½βˆ—οΏ½ οΏ½ 𝐴𝐴 βˆ— 𝐡𝐡 = οΏ½ 𝑔𝑔 β„Ž 𝑐𝑐 𝑑𝑑 (π‘Žπ‘Ž βˆ— 𝑒𝑒) + (𝑏𝑏 βˆ— 𝑔𝑔) (π‘Žπ‘Ž βˆ— 𝑓𝑓) + (𝑏𝑏 βˆ— β„Ž) οΏ½ =οΏ½ (𝑐𝑐 βˆ— 𝑒𝑒) + (𝑑𝑑 βˆ— 𝑔𝑔) (𝑐𝑐 βˆ— 𝑓𝑓) + (𝑑𝑑 βˆ— β„Ž)

𝑐𝑐 𝐢𝐢𝐢𝐢 𝑔𝑔 οΏ½οΏ½ βˆ— οΏ½οΏ½ 𝑓𝑓 𝑗𝑗 βˆ’

= (π‘Žπ‘Ž βˆ— 𝑔𝑔) βˆ’ (𝑑𝑑 βˆ— 𝑗𝑗) οΏ½(π‘Žπ‘Ž βˆ— β„Ž) βˆ’ (𝑑𝑑 βˆ— π‘˜π‘˜) (π‘Žπ‘Ž βˆ— 𝑖𝑖) βˆ’ (𝑑𝑑 βˆ— 𝑙𝑙)

(𝑏𝑏 βˆ— 𝑔𝑔) βˆ’ (𝑒𝑒 βˆ— 𝑗𝑗) (𝑏𝑏 βˆ— β„Ž) βˆ’ (𝑒𝑒 βˆ— π‘˜π‘˜) (𝑏𝑏 βˆ— 𝑖𝑖) βˆ’ (𝑒𝑒 βˆ— 𝑙𝑙)

𝐿𝐿𝐿𝐿 π‘Žπ‘Ž β€’ 𝐴𝐴 οΏ½ βˆ— οΏ½ 𝐡𝐡 = οΏ½ 𝑐𝑐 βˆ’

𝐿𝐿𝐿𝐿 𝑒𝑒 𝑏𝑏 οΏ½οΏ½ βˆ— οΏ½οΏ½ 𝑔𝑔 𝑑𝑑 βˆ’

β„Ž π‘˜π‘˜

𝑖𝑖 οΏ½ 𝑙𝑙

(𝑐𝑐 βˆ— 𝑔𝑔) βˆ’ (𝑓𝑓 βˆ— 𝑗𝑗) (𝑐𝑐 βˆ— β„Ž) βˆ’ (𝑓𝑓 βˆ— π‘˜π‘˜)οΏ½ (𝑐𝑐 βˆ— 𝑖𝑖) βˆ’ (𝑓𝑓 βˆ— 𝑙𝑙)

𝐢𝐢𝐢𝐢 π‘Žπ‘Ž 𝑏𝑏 𝑐𝑐 𝐢𝐢𝐢𝐢 𝑔𝑔 β„Ž 𝑖𝑖 οΏ½οΏ½ βˆ— οΏ½οΏ½ οΏ½ β€’ 𝐴𝐴 οΏ½ βˆ— οΏ½ 𝐡𝐡 = οΏ½ 𝑑𝑑 𝑒𝑒 𝑓𝑓 𝑗𝑗 π‘˜π‘˜ 𝑙𝑙 βˆ’ βˆ’ This operation can’t be done because the number of lines in the first matrix must be equal to the number of columns in the second matrix.

(π‘Žπ‘Ž βˆ— 𝑔𝑔) βˆ’ (𝑏𝑏 βˆ— β„Ž) βˆ’ (𝑐𝑐 βˆ— 𝑖𝑖) =οΏ½ (𝑑𝑑 βˆ— 𝑔𝑔) βˆ’ (𝑒𝑒 βˆ— β„Ž) βˆ’ (𝑓𝑓 βˆ— 𝑖𝑖)

𝑓𝑓 οΏ½ β„Ž

(π‘Žπ‘Ž βˆ— 𝑗𝑗) βˆ’ (𝑏𝑏 βˆ— π‘˜π‘˜) βˆ’ (𝑐𝑐 βˆ— 𝑙𝑙) οΏ½ (𝑑𝑑 βˆ— 𝑗𝑗) βˆ’ (𝑒𝑒 βˆ— π‘˜π‘˜) βˆ’ (𝑓𝑓 βˆ— 𝑙𝑙)

𝐿𝐿𝐿𝐿 𝐿𝐿𝐿𝐿 𝑒𝑒 𝑓𝑓 π‘Žπ‘Ž 𝑏𝑏 οΏ½οΏ½ βˆ— οΏ½οΏ½ οΏ½ β€’ 𝐴𝐴 οΏ½ βˆ— οΏ½ 𝐡𝐡 = οΏ½ 𝑔𝑔 β„Ž 𝑐𝑐 𝑑𝑑 βˆ’ βˆ’ This operation can’t be done because the number of columns in the first matrix must be equal to the number of lines in the second matrix.

2.1.3 The first operation is β€œ+” and the second is β€œ/” 𝐢𝐢𝐢𝐢 𝐢𝐢𝐢𝐢 𝑒𝑒 𝑓𝑓 π‘Žπ‘Ž 𝑏𝑏 οΏ½οΏ½ + οΏ½οΏ½ οΏ½ β€’ 𝐴𝐴 οΏ½ + οΏ½ 𝐡𝐡 = οΏ½ 𝑔𝑔 β„Ž 𝑐𝑐 𝑑𝑑 / / (π‘Žπ‘Ž + 𝑒𝑒)/(𝑐𝑐 + 𝑔𝑔) (𝑏𝑏 + 𝑒𝑒)/(𝑑𝑑 + 𝑔𝑔) οΏ½ =οΏ½ (π‘Žπ‘Ž + 𝑓𝑓)/(𝑐𝑐 + β„Ž) (𝑏𝑏 + 𝑓𝑓)/(𝑑𝑑 + β„Ž)

2.1.5 The first operation is β€œ*” and the second is β€œ-” with two rectangular matrices (2 columns and 3 lines) Suppose the following matrices: π‘Žπ‘Ž 𝑏𝑏 𝑔𝑔 β„Ž 𝐴𝐴 = οΏ½ 𝑐𝑐 𝑑𝑑 οΏ½ and 𝐡𝐡 = οΏ½ 𝑖𝑖 𝑗𝑗 οΏ½ 𝑒𝑒 𝑓𝑓 π‘˜π‘˜ 𝑙𝑙

𝐢𝐢𝐢𝐢 𝐢𝐢𝐢𝐢 π‘Žπ‘Ž 𝑏𝑏 + 𝑒𝑒 𝑓𝑓 οΏ½οΏ½ οΏ½οΏ½ οΏ½ β€’ 𝐴𝐴 οΏ½ + οΏ½ 𝐡𝐡 = οΏ½ 𝑔𝑔 β„Ž 𝑐𝑐 𝑑𝑑 / / (π‘Žπ‘Ž + 𝑒𝑒)/(𝑐𝑐 + 𝑓𝑓) (𝑏𝑏 + 𝑒𝑒)/(𝑑𝑑 + 𝑓𝑓) οΏ½ =οΏ½ (π‘Žπ‘Ž + 𝑔𝑔)/(𝑐𝑐 + β„Ž) (𝑏𝑏 + 𝑔𝑔)/(𝑑𝑑 + β„Ž)

ISBN: 978-1-61804-117-3

𝑏𝑏 𝑒𝑒

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π‘Žπ‘Ž 𝐢𝐢𝐢𝐢 β€’ 𝐴𝐴 οΏ½ βˆ— οΏ½ 𝐡𝐡 = οΏ½ 𝑐𝑐 𝑒𝑒 βˆ’

𝑏𝑏 𝐢𝐢𝐢𝐢 𝑔𝑔 𝑑𝑑 οΏ½ οΏ½ βˆ— οΏ½ οΏ½ 𝑖𝑖 𝑓𝑓 βˆ’ π‘˜π‘˜

= (π‘Žπ‘Ž βˆ— 𝑔𝑔) βˆ’ (𝑐𝑐 βˆ— 𝑖𝑖) βˆ’ (𝑒𝑒 βˆ— π‘˜π‘˜) οΏ½ (π‘Žπ‘Ž βˆ— β„Ž) βˆ’ (𝑐𝑐 βˆ— 𝑗𝑗) βˆ’ (𝑒𝑒 βˆ— 𝑙𝑙)

𝐢𝐢𝐢𝐢 β€’ �𝑂𝑂𝑂𝑂1 οΏ½ 𝐡𝐡𝑗𝑗𝑙𝑙 = 𝑀𝑀𝑖𝑖𝑙𝑙 (3) 𝑂𝑂𝑂𝑂2 With -𝑗𝑗 is the number of lines for the first matrix and it is equal to the number of columns for the second matrix. The number of lines for the first matrix must be equal to the number of columns in the second matrix when the operation between the two matrices is 𝐢𝐢𝐢𝐢 columns of the first matrix by the lines of the second matrix. - 𝑖𝑖 is the number of columns for the first matrix 𝐴𝐴. -𝑙𝑙 is the number of columns for the second matrix 𝐡𝐡. The result matrix has the number of lines equal to 𝑙𝑙 and the number of columns equal to 𝑖𝑖. 𝑗𝑗 𝐴𝐴𝑖𝑖

β„Ž 𝑗𝑗 οΏ½ 𝑙𝑙

(𝑏𝑏 βˆ— 𝑔𝑔) βˆ’ (𝑑𝑑 βˆ— 𝑖𝑖) βˆ’ (𝑓𝑓 βˆ— π‘˜π‘˜) οΏ½ (𝑏𝑏 βˆ— β„Ž) βˆ’ (𝑑𝑑 βˆ— 𝑗𝑗) βˆ’ (𝑓𝑓 βˆ— 𝑙𝑙)

π‘Žπ‘Ž 𝑏𝑏 𝐢𝐢𝐢𝐢 𝑔𝑔 β„Ž 𝐢𝐢𝐢𝐢 β€’ 𝐴𝐴 οΏ½ βˆ— οΏ½ 𝐡𝐡 = οΏ½ 𝑐𝑐 𝑑𝑑 οΏ½ οΏ½ βˆ— οΏ½ οΏ½ 𝑖𝑖 𝑗𝑗 οΏ½ 𝑒𝑒 𝑓𝑓 βˆ’ π‘˜π‘˜ 𝑙𝑙 βˆ’ This operation can’t be done because the number of lines in the first matrix must be equal to the number of columns in the second matrix. π‘Žπ‘Ž 𝐿𝐿𝐿𝐿 β€’ 𝐴𝐴 οΏ½ βˆ— οΏ½ 𝐡𝐡 = οΏ½ 𝑐𝑐 𝑒𝑒 βˆ’

(π‘Žπ‘Ž βˆ— 𝑔𝑔) βˆ’ (𝑏𝑏 βˆ— β„Ž) = οΏ½(𝑐𝑐 βˆ— 𝑔𝑔) βˆ’ (𝑑𝑑 βˆ— β„Ž) (𝑒𝑒 βˆ— 𝑔𝑔) βˆ’ (𝑓𝑓 βˆ— β„Ž)

𝑏𝑏 𝐿𝐿𝐿𝐿 𝑔𝑔 𝑑𝑑 οΏ½ οΏ½ βˆ— οΏ½ οΏ½ 𝑖𝑖 𝑓𝑓 βˆ’ π‘˜π‘˜

β„Ž 𝑗𝑗 οΏ½ 𝑙𝑙

(π‘Žπ‘Ž βˆ— 𝑖𝑖) βˆ’ (𝑏𝑏 βˆ— 𝑗𝑗) (𝑐𝑐 βˆ— 𝑖𝑖) βˆ’ (𝑑𝑑 βˆ— 𝑗𝑗) (𝑒𝑒 βˆ— 𝑖𝑖) βˆ’ (𝑓𝑓 βˆ— 𝑗𝑗)

(π‘Žπ‘Ž βˆ— π‘˜π‘˜) βˆ’ (𝑏𝑏 βˆ— 𝑙𝑙) (𝑒𝑒 βˆ— π‘˜π‘˜) βˆ’ (𝑑𝑑 βˆ— 𝑙𝑙)οΏ½ (𝑒𝑒 βˆ— π‘˜π‘˜) βˆ’ (𝑓𝑓 βˆ— 𝑙𝑙)

π‘Žπ‘Ž 𝑏𝑏 𝐿𝐿𝐿𝐿 𝑔𝑔 β„Ž 𝐿𝐿𝐿𝐿 β€’ 𝐴𝐴 οΏ½ βˆ— οΏ½ 𝐡𝐡 = οΏ½ 𝑐𝑐 𝑑𝑑 οΏ½ οΏ½ βˆ— οΏ½ οΏ½ 𝑖𝑖 𝑗𝑗 οΏ½ 𝑒𝑒 𝑓𝑓 βˆ’ π‘˜π‘˜ 𝑙𝑙 βˆ’ This operation can’t be done because the number of columns in the first matrix must be equal to the number of lines in the second matrix.

2.2 General Case of operation between matrices

𝐢𝐢𝐢𝐢 𝑗𝑗 𝑂𝑂𝑂𝑂 β€’ οΏ½ 1 οΏ½ 𝐡𝐡𝑙𝑙 = 𝑀𝑀𝑖𝑖𝑙𝑙 (2) 𝑂𝑂𝑂𝑂2 With -𝑗𝑗 is the number of lines for the two matrices. The number of lines for the two matrices should be equal when the operation between the two matrices is 𝐢𝐢𝐢𝐢 columns of the first matrix by the columns of the second matrix. - 𝑖𝑖 is the number of columns for the first matrix 𝐴𝐴. -𝑙𝑙 is the number of columns for the second matrix 𝐡𝐡. The result matrix has the number of lines equal to 𝑙𝑙 and the number of columns equal to 𝑖𝑖. 𝑗𝑗 𝐴𝐴𝑖𝑖

Example: 𝐢𝐢𝐢𝐢 𝐴𝐴12 �𝑂𝑂𝑂𝑂1 οΏ½ 𝐡𝐡31 = [β€’ 𝑂𝑂𝑂𝑂2 β€’ β€’ οΏ½β€’ β€’οΏ½ = 𝑀𝑀32 β€’ β€’

𝐢𝐢𝐢𝐢 β€’] �𝑂𝑂𝑂𝑂1 οΏ½ [β€’ 𝑂𝑂𝑂𝑂2

ISBN: 978-1-61804-117-3

β€’

Example: 𝐢𝐢𝐢𝐢 1 𝑂𝑂𝑂𝑂 𝐴𝐴2 οΏ½ 1 οΏ½ 𝐡𝐡13 = [β€’ 𝑂𝑂𝑂𝑂2

𝐢𝐢𝐢𝐢 β€’ β€’ β€’] �𝑂𝑂𝑂𝑂1 οΏ½ οΏ½β€’οΏ½ = οΏ½β€’ β€’ 𝑂𝑂𝑂𝑂2 β€’

β€’ β€’οΏ½ = 𝑀𝑀23 β€’

𝐿𝐿𝐿𝐿 𝑗𝑗 𝑗𝑗 β€’ 𝐴𝐴𝑖𝑖 �𝑂𝑂𝑂𝑂1 οΏ½ 𝐡𝐡𝑖𝑖𝑙𝑙 = 𝑀𝑀𝑙𝑙 (4) 𝑂𝑂𝑂𝑂2 With -𝑗𝑗 is the number of lines for the first matrix. - 𝑖𝑖 is the number of columns for the first matrix 𝐴𝐴. The number of columns for the first matrix must be equal to the number of columns in the second matrix when the operation between the two matrices is 𝐿𝐿𝐿𝐿 Lines of the first matrix by the lines of the second matrix. -𝑙𝑙 is the number of lines for the second matrix 𝐡𝐡. The result matrix has the number of lines equal to 𝑗𝑗 and the number of columns equal to 𝑙𝑙. Example: 𝐿𝐿𝐿𝐿 𝐴𝐴13 �𝑂𝑂𝑂𝑂1 οΏ½ 𝐡𝐡32 = [β€’ 𝑂𝑂𝑂𝑂2 [β€’ β€’] = 𝑀𝑀21

β€’

𝐿𝐿𝐿𝐿 β€’ β€’] �𝑂𝑂𝑂𝑂1 οΏ½ οΏ½ β€’ 𝑂𝑂𝑂𝑂2

β€’ β€’

β€’ οΏ½= β€’

𝐿𝐿𝐿𝐿 𝑗𝑗 𝑗𝑗 β€’ 𝐴𝐴𝑖𝑖 �𝑂𝑂𝑂𝑂1 οΏ½ 𝐡𝐡𝑙𝑙𝑖𝑖 = 𝑀𝑀𝑙𝑙 (5) 𝑂𝑂𝑂𝑂2 With -𝑗𝑗 is the number of lines for the first matrix. -𝑖𝑖 is the number of columns for the first matrix 𝐴𝐴. The number of columns for the first matrix must be equal to the number of lines in the second matrix when the operation between the two matrices is 𝐿𝐿𝐿𝐿 Lines of the first matrix by the columns of the second matrix. -𝑙𝑙 is the number of columns for the second matrix 𝐡𝐡.

β€’] =

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The result matrix has the number of lines equal to 𝑗𝑗 and the number of columns equal to 𝑙𝑙. Example: 𝐿𝐿𝐿𝐿 β€’ 𝐿𝐿𝐿𝐿 3 𝑂𝑂𝑂𝑂 1 𝐴𝐴1 οΏ½ 1 οΏ½ 𝐡𝐡3 = οΏ½β€’οΏ½ �𝑂𝑂𝑂𝑂1 οΏ½ [β€’ β€’ 𝑂𝑂𝑂𝑂2 𝑂𝑂𝑝𝑝2 β€’ β€’ β€’ οΏ½β€’ β€’ β€’οΏ½ = 𝑀𝑀33 β€’ β€’ β€’

β€’

about the importance of this new method of operation between matrices.

β€’ Example 1

β€’] =

Suppose we have three polynomial equations of the second order as following: 𝑦𝑦1 = π‘Žπ‘Žπ‘₯π‘₯ 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐 𝑦𝑦2 = 𝑑𝑑π‘₯π‘₯ 2 + 𝑒𝑒𝑒𝑒 + 𝑓𝑓 𝑦𝑦3 = 𝑔𝑔π‘₯π‘₯ 2 + β„Žπ‘₯π‘₯ + 𝑖𝑖 We want to put them into matrices in order to simplify the writing. Therefore,

2.3 The Case when we have only the first operator π‘Άπ‘Άπ‘Άπ‘ΆπŸπŸ

𝑦𝑦1 π‘Žπ‘Ž 𝑦𝑦 𝑑𝑑 οΏ½ 2οΏ½ = οΏ½ 𝑦𝑦3 𝑔𝑔

When we have only one operator we consider it as the first operator. In this case the two matrices must 𝑗𝑗 𝑗𝑗 have the same dimensions for example 𝐴𝐴𝑖𝑖 and 𝐡𝐡𝑖𝑖 with 𝑗𝑗 is the number of lines and 𝑖𝑖 is the number of columns for both matrices. The operation will be only between the elements of two matrices which are located in the same position.

𝑐𝑐 𝐿𝐿𝐿𝐿 π‘₯π‘₯ 2 𝑓𝑓� οΏ½ βˆ— οΏ½ οΏ½ π‘₯π‘₯ οΏ½ 𝑖𝑖 + 1

It can be written also in the following form π‘Žπ‘Ž 𝑑𝑑 𝑔𝑔 𝐢𝐢𝐢𝐢 π‘₯π‘₯ 2 [𝑦𝑦1 𝑦𝑦2 𝑦𝑦3 ] = �𝑏𝑏 𝑒𝑒 β„Ž οΏ½ οΏ½ βˆ— οΏ½ οΏ½ π‘₯π‘₯ οΏ½ 𝑐𝑐 𝑓𝑓 𝑖𝑖 + 1

Suppose the following matrices: 𝑒𝑒 𝑓𝑓 π‘Žπ‘Ž 𝑏𝑏 οΏ½ and 𝐡𝐡 = οΏ½ οΏ½ 𝐴𝐴 = οΏ½ 𝑔𝑔 β„Ž 𝑐𝑐 𝑑𝑑

β€’ Example 2

Suppose we have four polynomial equations as following: π‘₯π‘₯ 𝑦𝑦1 =

π‘Žπ‘Ž 𝑏𝑏 (𝑂𝑂𝑂𝑂 ) 𝑒𝑒 𝑓𝑓 οΏ½ οΏ½ 1 οΏ½ 𝑔𝑔 β„Ž 𝑐𝑐 𝑑𝑑 π‘Žπ‘Ž 𝑂𝑂𝑂𝑂1 𝑒𝑒 𝑏𝑏 𝑂𝑂𝑂𝑂1 𝑓𝑓 οΏ½ 𝐴𝐴(𝑂𝑂𝑂𝑂1 )𝐡𝐡 = οΏ½ 𝑐𝑐 𝑂𝑂𝑂𝑂1 𝑔𝑔 𝑑𝑑 𝑂𝑂𝑂𝑂1 β„Ž For example: consider that the first Operation (𝑂𝑂𝑂𝑂1 ) is replaced respectively by (+, *, -, /). Therefore the operation of two matrices will be as following: β€’ 𝐴𝐴(𝑂𝑂𝑂𝑂1 )𝐡𝐡 = οΏ½

(π‘₯π‘₯βˆ’1)

1

𝑦𝑦2 = π‘₯π‘₯(1 + ) π‘₯π‘₯ 𝑦𝑦3 = π‘₯π‘₯ 2 𝑦𝑦4 = π‘₯π‘₯(4π‘₯π‘₯ + 3) We want to put them into matrices in order to simplify the writing. Therefore,

Suppose the following matrices: 𝑒𝑒 𝑓𝑓 π‘Žπ‘Ž 𝑏𝑏 οΏ½ and 𝐡𝐡 = οΏ½ οΏ½ thus 𝐴𝐴 = οΏ½ 𝑔𝑔 β„Ž 𝑐𝑐 𝑑𝑑 π‘Žπ‘Ž + 𝑒𝑒 𝑏𝑏 + 𝑓𝑓 οΏ½ 𝐴𝐴(+)𝐡𝐡 = οΏ½ 𝑐𝑐 + 𝑔𝑔 𝑑𝑑 + β„Ž π‘Žπ‘Ž βˆ— 𝑒𝑒 𝑏𝑏 βˆ— 𝑓𝑓 οΏ½ 𝐴𝐴(βˆ—)𝐡𝐡 = οΏ½ 𝑐𝑐 βˆ— 𝑔𝑔 𝑑𝑑 βˆ— β„Ž π‘Žπ‘Ž βˆ’ 𝑒𝑒 𝑏𝑏 βˆ’ 𝑓𝑓 οΏ½ 𝐴𝐴(βˆ’)𝐡𝐡 = οΏ½ 𝑐𝑐 βˆ’ 𝑔𝑔 𝑑𝑑 βˆ’ β„Ž π‘Žπ‘Ž/𝑒𝑒 𝑏𝑏/𝑓𝑓 οΏ½ 𝐴𝐴(/)𝐡𝐡 = οΏ½ 𝑐𝑐/𝑔𝑔 𝑑𝑑/β„Ž

𝑦𝑦1 �𝑦𝑦

3

𝑦𝑦2 π‘₯π‘₯ 𝑦𝑦4 οΏ½ = οΏ½π‘₯π‘₯

β€’ Example 3

1 π‘₯π‘₯ οΏ½ (βˆ—) οΏ½π‘₯π‘₯βˆ’1 π‘₯π‘₯ π‘₯π‘₯

1+

1

π‘₯π‘₯

4π‘₯π‘₯ + 3

οΏ½

Suppose we have four polynomial equations as following: 𝑦𝑦1 = (π‘₯π‘₯ 2 + π‘₯π‘₯)(π‘₯π‘₯ + 3) 1 𝑦𝑦2 = οΏ½π‘₯π‘₯ 2 + οΏ½ (π‘₯π‘₯ + 2) π‘₯π‘₯ 𝑦𝑦3 = (π‘₯π‘₯ 3 + π‘₯π‘₯)(2 + 3) 1

𝑦𝑦4 = οΏ½π‘₯π‘₯ 3 + οΏ½ (2 + 2)

Remark: The operation between two matrices 𝐴𝐴(βˆ—)𝐡𝐡 is not the same as the traditional operation 𝐴𝐴 βˆ— 𝐡𝐡.

π‘₯π‘₯

We want to put them into matrices in order to simplify the writing. Therefore, 𝐿𝐿𝐿𝐿 π‘₯π‘₯ 3 2 𝑦𝑦1 𝑦𝑦2 �𝑦𝑦 𝑦𝑦 οΏ½ = οΏ½π‘₯π‘₯ 3 π‘₯π‘₯ οΏ½ οΏ½ + οΏ½ οΏ½ 1 2οΏ½ 3 4 π‘₯π‘₯ 2 π‘₯π‘₯ βˆ—

3 Practical examples of the General operation of matrices In this section we are going to see some practical examples in mathematics in order to give an idea

ISBN: 978-1-61804-117-3

𝑏𝑏 𝑒𝑒 β„Ž

33

Advances in Mathematical and Computational Methods

β€’ Example 4

first operation between two elements of the two matrices, with (𝑂𝑂𝑂𝑂1 ) can be replaced by an operator (+, -, *, /), the second operator (𝑂𝑂𝑂𝑂2 ) is the operator between the operated elements such as (π‘Žπ‘Ž 𝑂𝑂𝑂𝑂1 𝑏𝑏)𝑂𝑂𝑂𝑂2 (𝑐𝑐 𝑂𝑂𝑂𝑂1 𝑑𝑑). With (𝑂𝑂𝑂𝑂2 ) can be replaced by an operator (+, -, *, /). Many studies will follow this paper in order to find other applications in many domains as in science, engineering and mathematics.

Suppose we have three polynomial equations of the second order as following: 𝑦𝑦1 = π‘Žπ‘Žπ‘₯π‘₯ 3 + 𝑏𝑏π‘₯π‘₯ 2 + 𝑐𝑐 It can be written in the following form π‘Žπ‘Ž 𝐢𝐢𝐢𝐢 π‘₯π‘₯ 3 [𝑦𝑦1 ] = �𝑏𝑏 οΏ½ οΏ½ βˆ— οΏ½ οΏ½π‘₯π‘₯ 2 οΏ½ 𝑐𝑐 + 1

It can be written also in the following form π‘Žπ‘Ž 𝐢𝐢𝐢𝐢 [𝑦𝑦1 ] = �𝑏𝑏 οΏ½ οΏ½ βˆ— οΏ½ [π‘₯π‘₯ 3 π‘₯π‘₯ 2 1] 𝑐𝑐 +

References: [1] Arnold Vladimir I.; Cooke Roger, β€œOrdinary differential equations”, Berlin, DE; New York, NY:Springer-Verlag, ISBN 978-3-540-54813-3 (1992). [2] Artin Michael, β€œAlgebra”, Prentice Hall, ISBN 978-0-89871-510-1, (1991). [3] Association for Computing Machinery, β€œComputer Graphics”, Tata McGraw–Hill, ISBN 978-0-07-059376-3, (1979). [4] Baker Andrew J., β€œMatrix Groups: An Introduction to Lie Group Theory”, Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-185233-470-3, (2003). [5] Bau III David, Trefethen Lloyd N., β€œNumerical linear algebra”, Philadelphia, PA: Society for Industrial and Applied Mathematics, ISBN 978-089871-361-9, (1997). [6] Bretscher Otto, β€œLinear Algebra with Applications” (3rd ed.), Prentice Hall, (2005). [7] Bronson Richard, β€œSchaum's outline of theory and problems of matrix operations”, New York: McGraw–Hill, ISBN 978-0-07-007978-6, (1989). [8] Brown William A., β€œMatrices and vector spaces”, New York, NY: M. Dekker, ISBN 978-0-82478419-5, (1991). [9] Coburn Nathaniel, β€œVector and tensor analysis”, New York, NY: Macmillan, OCLC 1029828, (1955). [10] Conrey J. Brian, β€œRanks of elliptic curves and random matrix theory”, Cambridge University Press, ISBN 978-0-521-69964-8, (2007). [11] Gilbarg David, Trudinger Neil S., β€œElliptic partial differential equations of second order” (2nd ed.), Berlin, DE; New York, NY: SpringerVerlag, ISBN 978-3-540-41160-4, (2001).

β€’ Example 5

Suppose we have four polynomial equations as following: 𝑦𝑦1 =

𝑦𝑦2 = 𝑦𝑦3 = 𝑦𝑦4 =

οΏ½π‘₯π‘₯ 2 +π‘₯π‘₯οΏ½

π‘₯π‘₯ +3 1 οΏ½π‘₯π‘₯ 2 + οΏ½ π‘₯π‘₯

π‘₯π‘₯+2 οΏ½π‘₯π‘₯ 3 +π‘₯π‘₯οΏ½

2+3 1 οΏ½π‘₯π‘₯ 3 + οΏ½ π‘₯π‘₯

2+2

We want to put them into matrices in order to simplify the writing. Therefore, 𝐿𝐿𝐿𝐿 π‘₯π‘₯ 3 2 𝑦𝑦1 𝑦𝑦2 π‘₯π‘₯ π‘₯π‘₯ �𝑦𝑦 𝑦𝑦 οΏ½ = οΏ½ 3 οΏ½ οΏ½ + οΏ½ οΏ½ 1 2οΏ½ 3 4 π‘₯π‘₯ 2 / π‘₯π‘₯

4 Conclusion In this paper, the author introduced a new and original way of operation between matrices. The traditional operation between matrices is just a part of the general operation of matrices. The general operation of matrices is introduced in order to facilitate the writing of many complicated expressions into matrices and to facilitate the operation between matrices as we want. We can operate the Columns of the first matrix by the columns of the second matrix, or the columns of the first matrix by the Lines (rows) of the second matrix, or Lines of the first matrix by the line of the second matrix, or the Lines of the first matrix by the columns of the second matrix. This is not the case of the traditional operation between matrices. Mainly we have two operations between matrices, the first operation is between the elements of the two matrices and the second is the operation between the operated elements. For example: (π‘Žπ‘Ž (𝑂𝑂𝑂𝑂1 )𝑏𝑏) is the ISBN: 978-1-61804-117-3

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